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Wideband microwave measurements of the extinction cross section---Experimental
techniques
Larsson, Christer; Gustafsson, Mats; Kristensson, Gerhard
Published: 2009-01-01
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Larsson, C., Gustafsson, M., & Kristensson, G. (2009). Wideband microwave measurements of the extinction
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L
UNDUNI
VERS
I
TY
PO Box117
22100L
und
+46462220000
CODEN:LUTEDX/(TEAT-7182)/1-22/(2009)
Wideband microwave measurements
of the extinction cross
section—Experimental techniques
Christer Larsson, Mats Gustafsson, and Gerhard Kristensson
Electromagnetic Theory
Department of Electrical and Information Technology
Lund University
Sweden
Mats Gustafsson, and Gerhard Kristensson
{Mats.Gustafsson,Gerhard.Kristensson}@eit.lth.se
Department of Electrical and Information Technology
Electromagnetic Theory
Lund University
P.O. Box 118
SE-221 00 Lund
Sweden
Christer Larsson
[email protected]
[email protected]
Saab Bofors Dynamics AB
SE-581 88 Linköping
Sweden
Department of Electrical and Information Technology
Electromagnetic Theory
Lund University
P.O. Box 118
SE-221 00 Lund
Sweden
Editor: Gerhard Kristensson
c Christer Larsson et al., Lund, September 9, 2009
1
Abstract
This paper describes the development of a method to determine the extinction
cross section for a very large bandwidth in the microwave region. The method
is based on measurements of the radar cross section in the forward direction
to calculate the extinction cross section for the frequency range [2.5, 38] GHz.
The method is applicable to samples of arbitrary shape and composition and
can also be used for polarimetric measurements. Coherent background subtraction is used to remove the contribution from antenna to antenna coupling.
Time domain gating is employed to increase the sensitivity. Dierent calibration methods are discussed. The eciency of the background subtraction and
the accuracy of the measurements are estimated.
1
Introduction
The motivation for this study is to develop an experimental method that can be
used to verify recent theoretical work, which shows that the extinction cross section
integrated over all frequencies is related to the static properties of the scatterer, see
Refs. 8, 20, 21. A method based on polarimetric measurements of the radar cross
section (RCS) in the forward direction is therefore developed in order to verify the
theoretical work. The optical theorem is used to calculate the extinction cross section
from the measured forward RCS for arbitrary polarization of the incident radiation.
The method described in this paper can be applied to samples of arbitrary shape
and composition. This can be compared to the previously reported method based
on monostatic RCS measurements which was restricted to thin planar surfaces at
normal incidence, see Refs 14, 21.
Forward RCS measurement ranges and measurements are less common than
monostatic RCS ranges and measurements. There are several probable reasons for
this. One reason is that most radar applications are monostatic. An important
advantage of forward scattering and other bistatic radar systems is that they can
be used to detect objects which have a low monostatic radar signature, see Ref. 23.
In fact, shaping, which is a common method to reduce the monostatic RCS, is
not an ecient method to reduce the forward RCS. The forward RCS is mainly
a function of geometric cross section area regardless of shape, when the target is
large compared to the wavelength, see Ref. 12. The forward RCS cannot be reduced
signicantly by radar absorbing materials either, see Ref. 18. The disadvantage of
forward scattering and other bistatic radar systems is that they are in general more
complicated than monostatic systems, e.g., instead of one location for the monostatic
system a bistatic system requires two locations and time synchronization between
the locations. Some applications that use scattering in the forward direction are
systems for ground target identication, see Ref. 3, and radar fences, see Ref. 23.
The relative lack of applications where forward scattering is used means that most
RCS measurement ranges are set up for monostatic measurements.
Another reason for the relative scarcity of forward RCS ranges and measurements
is that forward RCS measurements in the laboratory are technically complicated
due to the large direct contribution from the transmitting antenna to the received
2
signal. The direct contribution to the signal has to be removed using coherent
subtraction. This puts considerable demands on the dynamic range and the stability
of the system.
There are only a few measurement ranges designed for forward RCS measurements described in the open literature. A system where the RCS in the forward
direction can be measured is described in Ref. 4. That system operates in the
[2, 12.4] GHz range with a measurement accuracy of 1 dB at a level of 18 dBsm for
the forward RCS.
An anechoic chamber, originally designed for monostatic RCS measurements, is
rearranged for polarimetric forward RCS measurements in the work described in the
present paper. The measurement setup is described and two dierent calibration
methods that can be used for the forward scattering measurements are evaluated.
The eciency of the data processing methods that are used, coherent background
subtraction and time domain gating, are determined and discussed. The accuracy
of the measured forward RCS and the extinction cross section is estimated by comparing with calculations.
2
Theory
Consider the direct scattering problem of a time harmonic plane electromagnetic
wave impinging in the k̂-direction on a bounded scatterer. The bistatic RCS, σ(f, r̂),
in the r̂ direction at the frequency f is then dened as, see Ref. 11,
σ(f, r̂) = lim 4πr2
r→∞
|E s (r)|2
,
|E i |2
(2.1)
where E i denotes the incident electric eld, E s denotes the scattered electric eld
and r = |r| denotes the magnitude of the position vector r = rr̂ . Introduce the
complex-valued bistatic RCS amplitude, A(f, r̂), in the r̂ direction. A(f, r̂) preserves the phase information in the measurement and it is dened as,
√ E ∗ (r) −ikr
A(f, r̂) = lim 2 πr s
e
,
r→∞
|E i |
(2.2)
where it is assumed that the time convention is e−iωt for the time harmonic wave
and k = 2πf /c0 . c0 is the phase velocity of light in free space.
σ(f, r̂) and A(f, r̂) are then related by,
σ(f, r̂) = |A(f, r̂)|2 .
(2.3)
In practice, the two polarization components of A(f, r̂) are measured separately for
an incident wave that is transmitted with a dened polarization in most RCS measurements. This means that measured RCS data usually is presented as a function
of the transmitted and received polarizations used in the measurement, see Ref. 10.
As an example, σVH and AVH would be the notations for the recorded RCS and
RCS amplitude component, respectively, for vertical transmitting and horizontal
receiving linear polarizations.
3
Transmitting
antenna
3.40 m
3.40 m
Receiving
antenna
2.38 m
Figure 1:
The gure shows the experimental setup in the anechoic chamber.
The scattering cross section, σs (f ), is the average of the bistatic RCS over all
angles, i.e., integration over the unit sphere Ω, namely,
Z
1
σs (f ) =
σ(f, r̂) dΩ,
(2.4)
4π Ω
where dΩ denotes the surface element of the unit sphere. The extinction cross
section, σext (f ), sometimes also designated the total cross section, see Ref. 17, is
dened as
σext (f ) = σs (f ) + σa (f ),
(2.5)
where σa (f ) is the absorption cross section, which is a measure of the power absorbed
by the scatterer, see Ref. 2. The extinction cross section can be determined using
(2.5) by measuring the bistatic RCS at all angles and the absorption cross section.
However, a more straightforward method is to measure the RCS amplitude in the
forward direction, A(f, k̂), and use the optical theorem to determine σext (f ), see
Ref. 16,
c0
(2.6)
σext (f ) = √ Im Aii (f, k̂),
πf
where Aii (f, k̂) is the component of A(f, k̂) that is co-polarized with the incident
electric eld, E i , i.e.,
Ei
(2.7)
Aii (f, k̂) = A(f, k̂) ·
|E i |
The optical theorem thus makes it possible to determine the extinction cross section
by a measurement of the RCS amplitude in the forward direction.
3
Measurements
3.1 Near and fareld measurements
In an ideal RCS measurement a plane wave impinges on the target. However, this is
not the case in a practical measurement where a nite distance between antenna and
4
0.5
d/m
0.4
0.3
0.2
0.1
f/GHz
5
Figure 2:
in (3.1).
10
15
20
25
30
35
40
The largest transverse size of the object in order to fulll the criterion
target has to be used. With increasing distance the spherical wave becomes more
and more like a plane wave over the target. On the other hand, as the distance
increases so do practical problems and cost. The question then becomes: How large
distance is needed for the RCS measurement of a certain target and how much will
the chosen distance contribute to measurement error? This problem does not have
a general solution valid for all targets. Dierent analyses show, see Refs. 11, 13, 17,
that if the phase dierence over the target is less than π/8, then the error in the
RCS-measurement due to the spherical wavefront is less than 1 dB for most targets.
This requirement on the phase deviation over a target of size d gives the relation,
r>
2d2
,
λ
(3.1)
where λ is the wavelength of the incident radiation and r is the distance between
antenna and target. (3.1) is commonly referred to as the Far-eld criterion, see
Ref. 11. It is commonly used to determine the necessary antenna to target distance
for an RCS-measurement.
Rewriting (3.1) gives a restriction for the maximum size of the measured objects
that can be used in a measurement setup with a xed distance between antenna and
target.
r
rc0
d<
,
(3.2)
2f
where f is the frequency and c0 is the speed of light. The measurement system
geometry is shown in Figure 1. Using this measurement distance, r=3.40 m, and
frequency range of interest for this work, Figure 2 is obtained. Figure 2 shows
that measurements that are within the limits given by the far eld criterion can be
5
Receiving
Transmitting
antenna
antenna
Target measurement
Calibration measurement
Direct path measurement
rt
rr
rd
Figure 3:
The three dierent measurement setups that are considered.
performed on samples that are smaller than 0.16 m and 0.11 m for measurements up
to 18 and 38 GHz, respectively.
3.2 Calibration
Calibration is a crucial component in all RCS measurements. The quality of the
calibration and the accuracy inherent in the calibration procedure have large impact
on the quality of the calibrated data acquired from the measurement. It is therefore
useful to analyze the dierent methods that can be used in some detail.
Two calibration methods are considered in this section using dierent combinations of the measurement setups shown in Figure 3. The rst method is the
traditional calibration method used in most RCS measurements. It uses a target
with known RCS as a calibration target to calibrate the target measurement. The
second method that is described here does not require a calibration target. It uses
a direct path measurement of the signal to calibrate the target measurement. This
is a feature which can be useful for many setups. To the best of our knowledge, the
latter method is not in frequent use in the RCS community. The following discussion
on calibration is restricted to forward scattering, i.e., r̂ = k̂, but can be generalized
to other directions as well.
6
RCS/dBsm
Experiment
With
cal. obj.
W/oTheory
cal. obj.
Theory
-5
-15
r = 25 mm
r = 20 mm
r = 15 mm
-20
r = 10 mm
-10
-25
-30
-35
-40
5
10
15
f/GHz
20
Figure 4:
Comparison between results from forward RCS measurements on metal
spheres using the two calibration methods described here. The RCS obtained from
Mie series calculation is shown as a reference.
3.2.1
Calibration using a calibration target
The rst calibration method that is described here, calibration using a calibration
target, is the standard method for RCS calibration. It is described in detail in
a number of sources, see e.g., Refs. 5, 11. The main features of this method are
outlined in this section.
Starting with one of many formulations of the radar equation, see Ref. 19, an
RCS measurement can be described by,
Pr (f ) =
Pt (f )Gt (f ) σ(f ) Gr (f )λ2 1
,
4πrt2
4πrr2 4π L(f )
(3.3)
where Pr (f ) is the power received at the receiving antenna, Pt (f ) is the power
transmitted, Gt (f ) is the gain of the transmitting antenna, σ(f ) is the RCS for
the target, Gr (f ) is the gain of the receiving antenna, rt is the distance between
the transmitting antenna and the target to be measured, rr is the distance between
the target to be measured and the receiving antenna, λ is the wavelength of the
radiation and L(f ) is the system losses in cables etc. The power measures, the gains
and the loss are not normalized in this stage of the measurement.
If the system is kept constant without change of geometry and hardware setup,
only allowing the change between dierent targets, then all parameters in (3.3)
except σ(f ) can be replaced by a parameter that only depends on f , K(f ),
Pr (f ) = K(f )σ(f ).
(3.4)
7
´/dB
-30
-35
-40
-45
-50
-55
-60
-65
-70
36
32
24
18
12
GHz
GHz
GHz
GHz
GHz
6 GHz
t/h
2
4
6
8
10 12 14 16 18 20
Figure 5:
The gure shows the subtraction eciency as a function of time, in
hours, for 2∆f = 2 GHz wide frequency bands. The curves are labeled with the
center frequencies of the frequency bands.
This leads to
σ(f ) =
Pr (f )
.
K(f )
(3.5)
In practice, coherent radar measurements are performed to determine the separate
polarization components of the RCS amplitude, Aij ,
Aij (f ) = vcal (f )Vij,r (f ),
(3.6)
where vcal (f ) is a complex-valued quantity that is sometimes referred to as a calibration vector, and Vij,r (f ) is the complex-valued amplitude as measured at the
receiver. vcal (f ) has to be determined in a calibration measurement using a target
with known RCS amplitude, Aij,cal (f ).
vcal (f ) =
Aij,cal (f )
,
Vij,r,cal (f )
(3.7)
where Vij,r,cal (f ) is the amplitude at the receiver of the calibration target.
Metallic spheres are used as calibration objects for all the measurements that are
presented in this paper. Spheres are ideal calibration objects for several reasons, as
pointed out in Ref. 4. The RCS in the forward direction is easy to calculate to very
good accuracy using the Mie series, see Ref. 17. The symmetry of a sphere means
that it is straightforward to align it. A sphere has also a much larger forward RCS
than its monostatic RCS. In fact, a sphere's forward RCS is similar to the forward
RCS of a at plate with the same cross section area.
8
20
10
0
-10
-20
-30
-40
-50
-60
-70
RCS/dBsm
Measured backgr.
+ Backgr. subtr.
+ Time gated
5
10
15
20
25
30
f/GHz
35
Figure 6:
Measurement of the background. The gure shows calibrated raw data
from a background measurement, data after subtracting another background measurement, and after additional time gating.
3.2.2
Calibration without calibration target
In some cases, nding and utilizing an accurate calibration target is complicated.
Especially for bistatic geometries this can be a problem. It is therefore of interest
to nd a method that does not require a calibration target. Only a few descriptions
of a method that does not use a calibration target are found in the literature, which
indicates that it is not very common and certainly much less frequently used than
the calibration using a calibration target.
This alternative method is based on measuring the direct signal from transmitting to receiving antenna. In Ref. 1 this procedure is used to calibrate forward
scattering measurements from dierent terrain types. The basic details are outlined
in Ref. 6 where the method is used to calibrate the measurements in an outdoors
bistatic RCS range. The method is also described and discussed in great detail in
Ref. 24 and test measurements on simple targets are performed in Ref. 25.
The transmitting and receiving antennas are positioned facing each other so that
the direct path signal can be measured. The radar equation then becomes
Pd (f ) =
Pt (f )Gt (f ) Gr (f )λ2 1
,
4πrd2
4π L(f )
(3.8)
where Pd (f ) is the measured direct path power and rd is the distance between the
transmitting and receiving antennas. (3.8) can be rewritten in the following form
Pd (f )4πrd2
Gr (f )λ2 1
= Pt (f )Gt (f )
,
4π L(f )
(3.9)
9
RCS/dBsm
20
10
0
-10
Measured
+ Backgr. subtr.
+ Time gated
-20
-30
-40
5
10
15
20
25
30
f/GHz
35
Figure 7:
Measurement of the forward RCS of a metal sphere with 10 mm radius.
The gure shows calibrated raw data, data after background subtraction and the
data after time domain gating.
Combining (3.9) and (3.3) and rewriting, one obtains
σ(f ) =
4πrt2 rr2 Pr (f )
.
rd2 Pd (f )
(3.10)
(3.10) can be rewritten in terms of the measured calibrated RCS amplitude component Aij (f )
√
2 πrt rr Vij,r (f )
,
(3.11)
Aij (f ) =
rd
Vij,d (f )
where Vij,d (f ) is the complex-valued amplitude as measured at the receiver for the
direct path measurement.
(3.11) can be simplied if the measured object is placed at the midpoint between
the transmitting and receiving antennas, i.e., if rt = rr = rd /2,
√
πrd Vij,r (f )
Aij (f ) =
,
(3.12)
2 Vij,d (f )
where the calibration vector, vcal (f ), can be identied as
√
πrd
.
vcal (f ) =
2Vij,d (f )
(3.13)
10
^
2
jAijj/dB
3.3 ns
-20
-30
-40
-50
-60
-70
-80
-90
-100
-8
-6
-4
-2
0
2
4
6
t/ns
8
Figure 8:
Time domain plot of the measurement of the forward RCS for a 10 mm
radius metal sphere.
3.2.3
Comparison and discussion of calibration methods
A comparison between the two methods is made by performing forward RCS measurements on small metal spheres and calibrating the measured data with both calibration methods. The calibration object is a sphere with radius 6 cm as described
in Section 3.5.
The results of the comparison measurement is shown in Figure 4. The rst
calibration method using a calibration target gives a more accurate result. In fact,
the data calibrated in this way is so close to the theoretical results that the curve
can not be distinguished from the theoretical curve in the gure.
The second method using the direct path signal to calibrate the measured data
gives a rather good result with a maximum deviation from the theoretical curve
of 0.7 dB. At present, it is not clear what causes this discrepancy. However, this
method is attractive for practical reasons so it would be of interest to investigate
the source of the discrepancy.
The rst method, using a calibration target, is chosen for the measurements that
we present in this report because of the better accuracy.
3.3 Background subtraction
Software or hardware gating is used in a monostatic measurement to reduce the
contribution from echoes that are not in the measurement gate. The method can
be used to remove echoes from the back wall and the antenna to antenna coupling.
However, it cannot be used to remove unwanted echoes that are at the same distance
from the radar as the target. For this reason coherent background subtraction is
commonly used to suppress background clutter in monostatic RCS measurements,
11
Figure 9:
Photograph of the anechoic chamber setup for forward RCS measurements. From left to right: transmitting antenna, EPS column for the sample to be
measured, and receiving antenna.
see Ref 9.
In a forward RCS experiment the situation is dierent. The antenna to antenna
coupling is then within the measurement gate in these measurements which means it
cannot be removed by time gating. Ecient coherent background subtraction then
becomes crucial to the success for a forward RCS measurement. The stability of the
system or rather, the lack of stability i.e., the drift with time will eectively limit
the eectiveness of the background subtraction.
Measurements are performed in order to investigate the long time stability of
the forward RCS measurement system. To do this consecutive background measurements of the forward scattering amplitude, Abg (f, k̂), are performed to determine
the eciency of the background subtraction as a function of time. Abg,0 (f, k̂) is
the forward scattering amplitude measured at t = t0 and Abg,n (f, k̂) is the forward
scattering amplitude measured at t = tn with tn > t0 . The subtraction eciency
is dened according to (3.14). Here η(f0 , tn ) is the subtraction eciency for the
frequency f = f0 at time t = tn . The eciency is obtained by averaging over a
frequency interval centered at f = f0 with m frequency points.
2
f0 +∆f 1 X Abg,n (f, k̂) − Abg,0 (f, k̂) η(f0 , tn ) =
m f =f −∆f Abg,n (f, k̂)
(3.14)
0
Figure 5 shows the subtraction eciency, η , as a function of time for six dierent,
2 GHz wide, frequency bands. The measurements are performed on two dierent
days. The bands with 6, 12 and 18 GHz center frequencies are measured at the
12
10 mm
15 mm
20 mm
25 mm
Figure 10:
The picture shows the metal spheres with dierent radii that are used
in the measurements.
same time and the bands with 24, 30 and 36 GHz center frequencies are measured
at the same time but on another day.
As expected and seen in the gure, the subtraction is most ecient at low frequencies. With reduced wavelength, the eect of displacements becomes larger. The
system losses, e.g., cable losses, also increase with increased frequency which means
a reduced signal to noise ratio (SNR). The reduced SNR reduces the eciency of
the subtraction. The gure shows that the background subtraction is most ecient
if the subtraction is performed within a few minutes relative to t0 . Focusing on
the rst 4 h of the measurements it is seen that the curves are not monotonically
increasing. Instead, the subtraction is more ecient at t = 4 h than at t = 2 h.
The cause of the reduction of the eciency of the background subtraction is interpreted as due to a change in distance between transmitting and receiving antenna.
This interpretation is conrmed by a corresponding phase shift in the background
measurement data. It is likely that this change in distance is caused by small
chamber contractions or expansions due to temperature changes. Estimates of the
temperature changes that are required to obtain these results can be made assuming
a 6.8 m antenna to antenna distance, a chamber steel support structure and a coefcient of linear expansion of 11.7 · 10−6 /◦ C for steel, see Ref. 15. The subtraction
eciency is 63 dB for the 6 GHz curve at t = 2 h and 30 dB for the 36 GHz curve at
t = 3 h. Assuming that the background subtraction eciency is limited by relative
displacements of the antennas, displacements of 5.6 µm and 24 µm are calculated
for 6 GHz and 36 GHz, respectively. Using the antenna to antenna distance and the
coecient for linear expansion for steel, the error levels correspond to temperature
changes of 0.07 ◦ C and 0.3 ◦ C, respectively. This means that it is extremely important to maintain a stable temperature if a good background subtraction eciency
is desired.
The result of a forward RCS measurement of the empty chamber is shown in
13
Experiment
Theory
RCS/dBsm
0
r = 25 mm
r = 20 mm
r = 15 mm
-10
r = 10 mm
-20
-30
-40
f/GHz
5
10
15
20
25
30
35
Figure 11:
Measured forward RCS for metal spheres with dierent radii. The
theoretical RCS determined from a Mie series calculation is also shown.
Figure 6. The gure also shows the results after coherent background subtraction
and time domain gating. The background subtraction suppresses the background
by more than 65 dB which gives a background level of less than 50 dBsm at the
lower frequencies. The background suppression becomes gradually less ecient with
increasing frequency and gives a background of less than 20 dBsm at the highest
frequencies. Time domain gating reduces the background by another 10 dB for the
lower frequencies and 15 dB for the higher frequencies.
The combined eect of coherent background subtraction and time domain gating
is then to suppress the background by 75 dB at low frequencies and by 50 dB at
high frequencies. This results in a remaining background level of 60 dBsm at low
frequencies and 35 dBsm at high frequencies.
3.4 Time domain gating
Coherent background subtraction reduces the level of the background for all measurement distances. However, strong returns can still remain in the background
after subtraction. Time domain gating, also denoted range gating can be used if
the unwanted return and the target return are separated in time, i.e., in range, see
Ref 9. The gating can be done in hardware, so called hardware gating, by pulsing
the RF signal from the radar. It can also be performed in software, so called software gating, by taking a discrete inverse fourier transform of the recorded frequency
domain data, Aij (f ).
Âij (t) = IFFT{Aij (f )}
(3.15)
14
¾ext/dBsm
Experiment
Theory
r = 25 mm
r = 20 mm
-25
r = 15 mm
-30
r = 10 mm
-35
-40
f/GHz
5
10
15
20
25
30
35
Figure 12:
Measured extinction cross section for metal spheres with dierent radii.
The theoretical extinction cross section determined from a Mie series calculation is
also shown.
A window is then applied to the resulting time domain data, Âij (t), and Âij,gat (t) is
obtained. The result is then transformed back to the frequency domain.
Aij,gat (f ) = FFT{Âij,gat (t)}
(3.16)
The results from a forward RCS measurement on a metal sphere with 10 mm
radius are shown in Figure 7. The gure also shows the result after coherent background subtraction and time domain gating.
Figure 8 shows the time domain data and the extent of the gate that was used
to obtain the gated data in in Figure 7.
Dierent window types, both Hanning and rectangular, are tested but the results
are very similar for both windows. It should also be noted here that it is important
that the window is suciently large so that it contains the entire response of the
target.
3.5 Experimental setup
This section sums up the experimental details and the measurement results presented
in this report.
The forward RCS measurements are performed in an anechoic chamber, see
Figures 1 and 9. A pair of wideband ridged horn antennas are positioned facing each
other at a distance of 6.80 m with the sample placed on an expanded polystyrene
(EPS) column at the midpoint. The measurement uses a Performance Network
Analyzer (PNA) transmitting a continuous wave without online hardware or software
gating. The source power from the PNA is set to 1 mW for the measurements
described here. The power output from the antenna is signicantly lower mainly
15
RCS/dBsm
20
10
HH backgr.
HV backgr.
0
-10
-20
-30
-40
2
4
6
8
10
12
14
16
18
f/GHz
20
Figure 13:
The gure shows the measured calibrated backgrounds for HH and HV
polarizations before background subtraction.
due to cable losses. Separate measurements using two dierent antenna pairs are
performed for the frequency intervals [1, 22] GHz and [16, 40] GHz.
Each interval is swept using 4000 frequency points. Eight such sweeps are averaged for each of the sample measurement, background measurement and calibration
measurement. The averaging serves to improve the signal to noise ratio.
The calibration is performed either with a 60 mm radius metal sphere for the
lower frequency range, [1, 22] GHz, or with a 30 mm radius metal sphere for the
higher frequency range, [16, 40] GHz. The calibration measurement with the sphere
is preceded and followed by measurements of the background that are averaged
and coherently subtracted from the calibration object measurement. The Mie series result for a perfectly electric conducting sphere, see Ref. 17, is divided by the
background subtracted calibration data to obtain a calibration vector.
The sample is then measured and the background is subtracted from the sample
data in the same way as for the calibration measurement. The background measurements are performed within less than 1 minute before/after each measurement
to minimize the inuence of the background.
The background subtracted sample data is nally calibrated and time domain
gated. The window size is chosen to minimize the inuence of the background but
containing the complete response from the target. In the measurements on the small
steel spheres a 3.3 ns gate is used. An 18 ns gate is used for the measurements on the
helix which has an extended response. The time domain ltering and a large antenna
beam width at low frequencies reduce the useful frequency range to [2.5, 38] GHz.
16
11.1
0.83
15.3
Figure 14:
gure.
4
Drawing of the helix sample. The dimensions, in mm, are shown in the
Results and discussion
Some measurements are performed to further investigate the performance of the
experimental setup. The results are compared with theoretical calculations. The
experimental results are used to determine the accuracy of the forward RCS measurements and the following determination of extinction cross section. Polarimetric
measurements are also performed by measuring for linear polarizations and combining the results to obtain circular polarizations.
4.1 Wideband measurements
Tests of the accuracy of the method to determine the forward RCS are performed
by measurements on metal high precision ball bearing spheres with 10 mm, 15 mm,
20 mm, and 25 mm radii for the [2.5, 38] GHz frequency range, see Fig 10.
The RCS is measured in the forward direction and the extinction cross section
is determined using the optical theorem, (2.6).
The results are shown in Figures 11 and 12. The forward RCS and the extinction
cross sections obtained from the Mie series are shown for reference in the gures.
The largest discrepancy between measurement and theory for the extinction cross
section is found for the 10 mm radius sphere where it is less than 0.5 dB at 31 dBsm.
4.2 Polarimetric measurements
The cross-polarization distortion in the transmit and receive channels can be estimated measuring the background signal using the setup in Figure 1 without a target.
Figure 13 shows the results from such a measurement of the calibrated background
17
RCS/dBsm
-20
-30
-40
HH
VV
HV
VH
-50
-60
f/GHz
-70
2
Figure 15:
nents.
4
6
8
10
12
14
16
18
20
The gure shows the measured linearly polarized forward RCS compo-
signal for the HH and HV components. The suppression, i.e., the dierence between
the two curves, varies from approximately 20 dB to more than 30 dB depending on
frequency.
The forward RCS of a helix, see Figure 14, is measured with the incident wave
direction, k̂, along the axis of the helix. A helix is chosen because it is expected
to give dierent extinction cross sections for dierent circular polarizations of the
incident wave. The four linearly polarized components, HH, VV, HV and VH, of
the forward RCS obtained from a measurement of the helix sample are shown in
Figure 15. The corresponding forward components of the RCS amplitude are used
to determine the circular polarized components. The LL and RR forward RCS
components are shown in Figures 16 and 17, respectively. The results are compared
with method-of-moments calculations to validate the method. It is found that the
measured and calculated results qualitatively agree well.
The corresponding forward RCS amplitude components can be used with (2.6) to
obtain extinction cross sections for circular polarized incident waves. The resulting
extinction cross sections for the helix for left circular polarized incident wave and for
a right circular polarized incident wave are shown in Figures 18 and 19, respectively.
There is also, as expected, a large dierence between the two polarizations due to the
geometry of the helix. The results are also here compared with method-of-moments
calculations.
Polarimetric calibration to reduce the eect of cross-polarization distortion in the
transmit and receive channels is considered for future development of the method
presented in this paper. One could e.g., use methods analogous to polarimetric
calibration methods suggested for monostatic measurements. One should then use
at least two calibration targets with signicant cross-polarization return and known
scattering matrix, see Ref. 22. A straightforward implementation is found in Ref. 7
where a dihedral reector is used for polarimetric calibration of monostatic RCS
18
RCS/dBsm
-20
-25
-30
-35
Measurement
Calculation
LL
-40
-45
-50
f/GHz
2
4
6
8
10
12
14
16
18
20
Figure 16:
The gure shows the forward RCS for LL-polarization determined from
experiments and calculations.
measurements. However, nding a calibration object for broadband forward scattering polarimetric calibration is more complicated than in the monostatic case.
5
Conclusions
A method to determine the extinction cross section utilizing forward RCS measurements is developed. It is used to determine the extinction cross section with good
accuracy for general objects for the frequency range [2.5, 38] GHz.
The method presented in this paper can be used for arbitrary polarization of the
incident radiation for targets smaller than 0.16 m up to 18 GHz and smaller than
0.11 m up to 38 GHz. It is found that a calibration using a calibration target is more
accurate than an alternative calibration method that does not require a calibration
target. It is concluded that the alternative method needs further analysis to be
useful for this purpose.
It is suggested that changes in the relative distance from transmitting antenna to
receiving antenna are the limiting factor for the background subtraction eciency.
Software time domain gating can be used to suppress echoes that do not emanate
from the target but care has to taken so that the entire target response is included
in the gate. It is found that the combined eect of coherent background subtraction
and time domain gating is to suppress the background by 75 dB at low frequencies
and by 50 dB at high frequencies. This results in a remaining background level of
60 dBsm at low frequencies and 35 dBsm at high frequencies for a forward RCS
measurement.
19
-20
RCS/dBsm
-25
-30
-35
-40
RR
Measurement
Calculation
-45
-50
f/GHz
Figure 17:
The gure shows the forward RCS for RR-polarization determined from
experiments and calculations.
The method is validated by measuring small metal spheres with known extinction
cross sections. It is determined that the extinction cross section can be measured
with good accuracy down to a level of 30 dBsm for the investigated frequency range.
Measurements of the RCS amplitude linear polarization components can be used to
calculate the extinction cross section for a circular polarized incident wave.
Acknowledgments
The nancial support by the Swedish Research Council, the Swedish Foundation for
Strategic Research and the Swedish Defense Materiel Administration is gratefully
acknowledged. We would also like to acknowledge the technical assistance from Mats
Andersson at Saab Bofors Dynamics.
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